Problem: Factor the following expression: $-7$ $x^2+$ $33$ $x+$ $10$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(10)} &=& -70 \\ {a} + {b} &=& & & {33} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-70$ and add them together. Remember, since $-70$ is negative, one of the factors must be negative. The factors that add up to ${33}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-2}$ and ${b}$ is ${35}$ $ \begin{eqnarray} {ab} &=& ({-2})({35}) &=& -70 \\ {a} + {b} &=& {-2} + {35} &=& 33 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 {-2}x +{35}x +{10} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 {-2}x) + ({35}x +{10}) $ Factor out the common factors: $ x(-7x - 2) - 5(-7x - 2) $ Notice how $(-7x - 2)$ has become a common factor. Factor this out to find the answer. $(-7x - 2)(x - 5)$